Snubber circuit and power conversion apparatus

ABSTRACT

Provided is a snubber circuit comprising N parallel charging paths each having a positive-side capacitor, a first diode, and a negative-side capacitor sequentially connected in series between a positive-side terminal and a negative-side terminal, and configured to conduct current from the positive-side terminal toward the negative-side terminal; (N+1) parallel discharging paths each having a second diode connected between the negative-side terminal or the negative-side capacitor of k th  charging path of N charging paths and the positive-side capacitor of (k+1) th  charging path of N charging paths or the positive-side terminal, and configured to counduct current from the negative-side terminal toward the positive-side terminal via at least one of the negative-side capacitor and the positive-side capacitor; and at least one auxiliary capacitor each being connected in parallel to at least one of the N first diodes included on N charging paths and (N+1) second diodes included on (N+1) discharging paths.

The contents of the following Japanese patent application (s) areincorporated herein by reference: 2019-156565 filed in JP on Aug. 29,2019.

BACKGROUND 1. Technical Field

The present invention relates to a snubber circuit and a powerconversion apparatus.

2. Related Art

In the related art, a variety of technologies for preventing elementbreakdown due to a surge voltage are suggested (for example, refer toPatent Document 1).

-   Patent Document 1: Japanese Patent Application Publication No.    2016-144340

In recent years, it is needed to effectively reduce the surge voltage,in association with an increase in current of a semiconductor module.

SUMMARY

In order to solve the above problem, a first aspect of the presentinvention provides a snubber circuit. The snubber circuit may comprise N(N: integer equal to or greater than 1) parallel charging paths eachhaving a positive-side capacitor, a first diode, and a negative-sidecapacitor sequentially connected in series between a positive-sideterminal and a negative-side terminal, and configured to cause currentto flow from a side of the positive-side terminal toward a side of thenegative-side terminal. The snubber circuit may comprise (N+1) paralleldischarging paths each having a second diode connected between thenegative-side terminal or the negative-side capacitor of a k^(th)charging path (k: integer equal to or greater than 0 and smaller than N)of the N charging paths and the positive-side capacitor of a (k+1)^(th)charging path of the N charging paths or the positive-side terminal, andconfigured to cause current to flow from the side of the negative-sideterminal toward the side of the positive-side terminal via at least oneof the negative-side capacitor and the positive-side capacitor. Thesnubber circuit may comprise at least one auxiliary capacitor each beingconnected in parallel to at least one of the N first diodes included onthe N charging paths and the (N+1) second diodes included on the (N+1)discharging paths.

A capacity of the auxiliary capacitor may be less than a capacity ofeach positive-side capacitor and a capacity of each negative-sidecapacitor.

The capacity of the auxiliary capacitor may be 1/1000 to 1/100 of thecapacity of each positive-side capacitor and the capacity of eachnegative-side capacitor.

Each auxiliary capacitor may be connected in parallel to any one of thefirst diodes and the second diode.

Each auxiliary capacitor may be connected in parallel to any one of eachof the N first diodes and each of the (N+1) second diodes.

A wire inductance of each charging path may be less than a wireinductance of each discharging path.

A second aspect of the present invention provides a power conversionapparatus. The power conversion apparatus may comprise the snubbercircuit of the first aspect. The power conversion apparatus may comprisea switch circuit connected to the positive-side terminal and thenegative-side terminal.

The switch circuit may be an inverter having upper and lower arms. Whenany one of the upper and lower arms becomes non-conductive, a period ΔT1after a voltage applied to the arm reaches a power supply voltage untilthe voltage becomes a summed voltage of the positive-side capacitor andthe negative-side capacitor in series with each other and a period ΔT 2from an end of the period ΔT1 to an end of charging of at least one ofthe positive-side capacitor and the negative-side capacitor may satisfythe following relationship: ΔT1 is equal to or smaller than ΔT2 and ΔT2is smaller than 5×ΔT1.

The summary clause does not necessarily describe all necessary featuresof the embodiments of the present invention. The present invention mayalso be a sub-combination of the features described above.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a circuit diagram of a power conversion apparatus 1 inaccordance with the present embodiment.

FIG. 2 shows flow of current when a switching element 11 is turned offin Comparative Example.

FIG. 3 shows flow of current when the switching element 11 is turned onin Comparative Example.

FIG. 4 shows flow of current in a mode (1).

FIG. 5 shows flow of current in a mode (2).

FIG. 6 shows flow of current in a mode (3).

FIG. 7 shows flow of current in a mode (4).

FIG. 8 shows flow of current in a mode (5).

FIG. 9 shows voltages applied to the switching element 11 when theswitching element 11 is turned off and becomes non-conductive.

FIG. 10 shows a power conversion apparatus 1A in accordance with amodified embodiment.

FIG. 11 shows flow of current in a mode (1A).

FIG. 12 shows flow of current in a mode (2A).

FIG. 13 shows flow of current in a mode (3A).

FIG. 14 shows flow of current in a mode (4A).

FIG. 15 shows flow of current in a mode (5A).

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, the present invention will be described through embodimentsof the invention. However, the embodiments do not limit the inventiondefined in the claims. Also, all combinations of features described inthe embodiments are not necessarily essential to solution means of theinvention.

1. Circuit Configuration of Power Conversion Apparatus

FIG. 1 is a circuit diagram of a power conversion apparatus 1 inaccordance with the present embodiment. The power conversion apparatus 1is a circuit for one phase configured to convert DC power into polyphaseAC power. The power conversion apparatus 1 outputs, from a power supplyoutput terminal 19, a voltage converted by switching connection betweeneach electrode of a power supply capacitor 10 and the power supplyoutput terminal 19. In the meantime, a return path of AC current to beoutput may be the power supply output terminal 19 for other phase. Thepower supply output terminal 19 may be connected to an inductive load(not shown). The power conversion apparatus 1 includes a power supplycapacitor 10, a switch circuit 3, and a snubber circuit 2. In themeantime, the power conversion apparatus 1 may convert the DC power intosingle phase AC power with the switch circuit 3. In this case, the powerconversion apparatus 1 may include two power supply capacitors 10connected in series, and the return path of AC current to be output fromthe power supply output terminal 19 may be a center point between thepower supply capacitors 10.

The power supply capacitor 10 functions as a DC power supply. Oneterminal of the power supply capacitor 10 is connected to apositive-side wire 101, and the other terminal is connected to anegative-side wire 102. In FIG. 1, one power supply capacitor 10 isshown. However, the power conversion apparatus 1 may include a pluralityof power supply capacitors 10 connected in series or in parallel.

The switch circuit 3 is connected between the positive-side wire 101 andthe negative-side wire 102. Thereby, the switch circuit 3 is connectedbetween a positive-side terminal 201 and a negative-side terminal 202 ofthe snubber circuit 2, which will be described later. The switch circuit3 of the present embodiment may be a DC/AC inverter, and includesswitching elements 11 and 12 as an upper arm and a lower arm in thepower conversion apparatus 1, and flywheel diodes 13 and 14.

The switching elements 11 and 12 are sequentially connected in seriesbetween the negative-side wire 102 and the positive-side wire 101. Adrain terminal and the source terminal of each of the switching elements11 and 12 are respectively connected to the side of the positive-sidewire 101 and the side of the negative-side wire 102. Gate terminals ofthe switching elements 11 and 12 are connected to a gate driving circuit(not shown), so that on and off operations of the switching elements 11and 12 are controlled. For example, the switching elements 11 and 12 maybe controlled so that they are to be selectively in a connection statewith dead time, for which both the switching elements are in off states,being provided therebetween. The switching elements 11 and 12 may becontrolled in a PWM manner. A center point between the switching element11 and the switching element 12 is connected to the power supply outputterminal 19.

The switching elements 11 and 12 may be silicon semiconductor elementswhose base material is silicon or wide band gap semiconductor elements.The wide band gap semiconductor element is a semiconductor elementhaving a greater bandgap than the silicon semiconductor element, and is,for example, a semiconductor element including SiC, GaN, diamond, agallium nitride-based material, a gallium oxide-based material, AlN,AlGaN, ZnO or the like. In the meantime, the switching elements 11 and12 may be MOSFETs or semiconductor elements having other structure, suchas IGBT, bipolar transistor and the like.

The flywheel diodes 13 and 14 are anti-parallel connected to theswitching elements 11 and 12 so that a side facing toward thepositive-side wire 101 is to be a cathode, respectively. The flywheeldiodes 13 and 14 may be Schottky barrier diodes. The flywheel diodes 13and 14 may be silicon semiconductor elements or wide band gapsemiconductor elements.

At least two of the switching elements 11 and 12 and the flywheel diodes13 and 14 may be modularized as a semiconductor module 5. In the presentembodiment, as an example, the switching elements 11 and 12 and theflywheel diodes 13 and 14 are modularized as the semiconductor module 5.In this case, the drain terminal of the positive-side switching element11 may be a positive-side terminal 51 of the semiconductor module 5, andthe source terminal of the negative-side switching element 12 may be anegative-side terminal 52 of the semiconductor module 5.

[1.1. Snubber Circuit 2]

The snubber circuit 2 protects each element of the power conversionapparatus 1 by absorbing a surge voltage that is generated when theswitching elements 11 and 12 interrupt current. The snubber circuit 2may be connected between the positive-side wire 101 and thenegative-side wire 102 via a positive-side terminal 201 and anegative-side terminal 202. In the meantime, a wire (as an example, awire including the positive-side wire 101 and the negative-side wire102) between the snubber circuit 2 and the power supply capacitor 10 mayhave a wire inductance 1011, in accordance with a wire length thereof.Also, a wire (as an example, a wire including the positive-side wire 101and the negative-side wire 102) between the snubber circuit 2 and theswitching elements 11 and 12 may have a wire inductance 1012, inaccordance with a wire length thereof.

The snubber circuit 2 includes N parallel charging paths 21, (N+1)parallel discharging paths 22, and at least one auxiliary capacitor 252.In the meantime, quantity N is an integer equal to or greater than 1,and is 3, for example, in the present embodiment. Also, in the presentembodiment, as an example, the three charging paths 21 are referred toas a first charging path 21(1), a second charging path 21(2) and a thirdcharging path 21(3), in corresponding order from the left side in thedrawings. Also, the four discharging paths 22 are referred to as a firstdischarging path 22(1), a second discharging path 22(2), a thirddischarging path 22(3), and a fourth discharging path 22(4), incorresponding order from the left side in the drawings.

Each charging path 21 has a positive-side capacitor 211, a first diode212, and a negative-side capacitor 213 sequentially connected in seriesbetween the positive-side terminal 201 and the negative-side terminal202. Each of the positive-side capacitor 211 and the negative-sidecapacitor 213 functions as a snubber capacitor, and may absorb aninstantaneous surge voltage that is generated when the switchingelements 11 and 12 are driven (as an example, a surge voltage to beapplied to an element during a period greater than 10 ns and less than10 μs). For example, the positive-side capacitor 211 and thenegative-side capacitor 213 may suppress vibrations higher than 100 kHzand lower than 100 MHz. The positive-side capacitor 211 and thenegative-side capacitor 213 may be film capacitors or stacked ceramiccapacitors, as an example.

The first diode 212 is arranged so that an anode faces toward the sideof the positive-side terminal 201 and a cathode faces toward the side ofthe negative-side terminal 202. Thereby, each charging path 21 causesthe current to flow from the side of the positive-side terminal 201toward the side of the negative-side terminal 202.

Each discharging path 22 has a second diode 221. The second diode 221 isconnected between the negative-side terminal 202 or the negative-sidecapacitor 213 of a k^(th) charging path 21 (k: integer equal to orgreater than 0 and equal to or smaller than N) of the N charging paths21 and the positive-side capacitor 211 of a (k+1)^(th) charging path 21of the N charging paths 21 or the positive-side terminal 201. Forexample, the second diode 221 of the first discharging path 22(1) isconnected between the negative-side terminal 202 and the positive-sidecapacitor 211 of the first charging path 21(1). The second diode 221 ofthe second discharging path 22(2) is connected between the negative-sidecapacitor 213 of the first charging path 21(1) and the positive-sidecapacitor 211 of the second charging path 21(2). The second diode 221 ofthe third discharging path 22(3) is connected between the negative-sidecapacitor 213 of the second charging path 21(2) and the positive-sidecapacitor 211 of the third charging path 21(3). The second diode 221 ofthe fourth discharging path 22(4) is connected between the negative-sidecapacitor 213 of the third charging path 21(3) and the positive-sideterminal 201. The second diode 221 is arranged so that an anode facestoward the side of the k^(th) charging path 21(k) or the negative-sideterminal 202 and a cathode faces toward the side of the (k+1)^(th)charging path 21(k+1) or the positive-side terminal 201. Thereby, eachdischarging path 22 causes the current to flow from the side of thenegative-side terminal 202 toward the side of the positive-side terminal201 via at least one of the negative-side capacitor 213 and thepositive-side capacitor 211.

In the meantime, a wire inductance of each charging path 21 may be lessthan a wire inductance of each discharging path 22. For example, a wirelength of each charging path 21 may be shorter than a wire length ofeach discharging path 22. As an example, the wire length of eachcharging path 21 connecting the positive-side terminal 201 and thenegative-side terminal 202 may be shorter than the wire length of eachdischarging path 22 connecting the positive-side terminal 201 and thenegative-side terminal 202.

The auxiliary capacitors 252 are each connected in parallel to at leastone of the (N+1) second diodes 221 included in the (N+1) dischargingpaths 22. In the present embodiment, as an example, the snubber circuit2 has the (N+1) auxiliary capacitors 252, and each of the auxiliarycapacitors 252 is connected in parallel to each of the (N+1) seconddiodes 221.

A capacity of each auxiliary capacitor 252 may be less than a capacityof each positive-side capacitor 211 and a capacity of each negative-sidecapacitor 213. For example, the capacity of the auxiliary capacitor 252may be 1/1000 to 1/100 of the capacity of each positive-side capacitor211 and the capacity of each negative-side capacitor 213. The capacitiesof each of the auxiliary capacitors 252 may be the same as or differentfrom each other.

Also, a charging voltage of each auxiliary capacitor 252 may be lowerthan a charging voltage of the negative-side capacitor 213 at a timingat which the switching elements 11 and 12 interrupt current. Thereby,each auxiliary capacitor 252 may draw current flowing from the side ofthe positive-side terminal 201 toward the first diode 212.

[1.1.1. Operations of Snubber Circuit 2]

[1.1.1(1). Operations of Snubber Circuit of Comparative Example]

Before describing operations of the snubber circuit 2 of the presentembodiment, operations of a snubber circuit 200 (refer to FIGS. 2 and 3)of Comparative Example are described. The snubber circuit 200 isdifferent from the snubber circuit 2, in that the auxiliary capacitor252 is not provided.

First, operations are described which are performed when the switchingelement 11 is turned off in a state in which the switching element 11 isin an on state and the switching element 12 is in an off state. In thestate in which the switching element 11 is in an on state and theswitching element 12 is in an off state, the output current flowsthrough a path of the power supply capacitor 10, the positive-side wire101, the switching element 11 and the power supply output terminal 19.At this time, the output current flows through the wire inductance 1012and energy is accumulated therein.

FIG. 2 shows flow of current when the switching element 11 is turned offfrom this state, in Comparative Example. In the meantime, the brokenline arrows in FIG. 2 indicate the flow of current, and the solid linearrows indicate voltages of the power supply capacitor 10, thepositive-side capacitor 211, the negative-side capacitor 213 and thelike, and voltages that are generated by the wire inductance 1012 andthe like.

When the switching element 11 is turned off, the output current iscommutated, so that it flows from the power supply capacitor 10 and thepositive-side wire 101 through the positive-side capacitor 211, thefirst diode 212 and the negative-side capacitor 213 of each chargingpath 21 and is output from the power supply output terminal 19 via theflywheel diode 14. Thereby, the current energy of the wire inductance1012 is absorbed by charging of the positive-side capacitor 211 andnegative-side capacitor 213 of the charging path 21. Then, the outputcurrent is all finally commutated to a path of the power supplycapacitor 10, the negative-side wire 102, the flywheel diode 14 and thepower supply output terminal 19. Thereby, the commutation associatedwith the turn-off operation of the switching element 11 is completed.

FIG. 3 shows flow of current when the switching element 11 is againturned on from the state in which the turn-off operation of theswitching element 11 is completed, in Comparative Example.

When the switching element 11 is again turned on, the output currentflowing through the path of the power supply capacitor 10, thenegative-side wire 102, the flywheel diode 14 and the power supplyoutput terminal 19 is commutated to a path of the power supply capacitor10, the negative-side wire 102, the second diode 221 of each dischargingpath 22, the switching element 11 and the power supply output terminal19. At this time, the energy during the turn-off operation, which isaccumulated in the positive-side capacitor 211 and/or the negative-sidecapacitor 213 provided on the anode-side/cathode-side of the seconddiode 221, is released. Then, the output current is all finallycommutated to the path of the power supply capacitor 10, thepositive-side wire 101, the switching element 11 and the power supplyoutput terminal 19. Thereby, the commutation associated with the turn-onoperation of the switching element 11 is completed.

Herein, the voltages of the positive-side capacitor 211 and thenegative-side capacitor 213 during the turn-off and turn-on operationsof the switching element 11 are described. During the turn-offoperation, a relationship between the voltages of the positive-sidecapacitor 211 and the negative-side capacitor 213 of each charging path21 is expressed by a following equation (1). In the equation, Eindicates a voltage of the power supply capacitor 10, and V_(dc-off)indicates a voltage between the terminals, i.e., a voltage between thepositive-side wire 101 and the negative-side terminal 202 during theturn-off operation. Also, V_(p(1)) to V_(p(3)) indicate voltages of thepositive-side capacitors 211 of the first charging path 21(1) to thethird charging path 21(3). Also, V_(n(1)) to V_(n(3)) indicate voltagesof the negative-side capacitors 213 of the first charging path 21(1) tothe third charging path 21(3).

$\begin{matrix}\begin{matrix}{{E \leq \left( {V_{p{(1)}} + V_{n{(1)}}} \right)} = \left( {V_{p{(2)}} + V_{n{(2)}}} \right)} \\{= \left( {V_{p{(3)}} + V_{n{(3)}}} \right)} \\{= V_{{dc}\text{-}{off}}}\end{matrix} & (1)\end{matrix}$

Also, during the turn-on operation, a relationship between the voltagesof the positive-side capacitor 211 and the negative-side capacitor 213of each charging path 21 is expressed by a following equation (2). Inthe equation, V_(dc-oN) indicates a voltage between the terminals, i.e.,a voltage between the positive-side wire 101 and the negative-sideterminal 202 during the turn-on operation.

$\begin{matrix}\begin{matrix}{{E \geq V_{p{(1)}}} = \left( {V_{n{(1)}} + V_{p{(2)}}} \right)} \\{= \left( {V_{n{(2)}} + V_{p{(3)}}} \right)} \\{= V_{n{(3)}}} \\{= V_{{dc}\text{-}{oN}}}\end{matrix} & (2)\end{matrix}$

By the equations (1) and (2), a relationship between voltages of eachpositive-side capacitor 211 and each negative-side capacitor 213 isexpressed by a following equation (3) (also, refer to the voltages shownin FIGS. 2 and 3). In the equation, V_(dc) indicates a voltage betweenthe terminals, i.e., a voltage between the positive-side terminal 51 andthe negative-side terminal 52 in a stationary state.

$\begin{matrix}\begin{matrix}{E = {V_{dc} \approx V_{p{(1)}}}} \\{= V_{n{(3)}}} \\{= {1.5 \times V_{p{(2)}}}} \\{= {1.5 \times V_{n{(2)}}}} \\{= {3 \times V_{n{(1)}}}} \\{= {3 \times V_{p{(3)}}}}\end{matrix} & (3)\end{matrix}$

From the equation (3), it can be seen that, when capacitor current isinterrupted, a charging voltage (as an example, 4E/3 in FIG. 3) of eachcharging path 21 is higher than a discharging voltage (as an example, Ein FIG. 3) of each discharging path 22. In the meantime, also in theturn-on and turn-off operations of the switching element 12 when theoutput current flows in a reverse direction, the similar effects areachieved due to symmetry of the circuit, and the detailed descriptionsthereof are thus omitted.

According to the snubber circuit 200 of Comparative Example, the Nparallel charging paths 21 each of which has the positive-side capacitor211 and the negative-side capacitor 213 are provided. Therefore, whenthe current is interrupted by the semiconductor module 5, the energyaccumulated in the wire inductance 1012 charges the positive-sidecapacitor 211 and the negative-side capacitor 213 to voltages higherthan the voltage between the positive-side wire 101 and thenegative-side terminal 202, through each charging path 21. Thereby,element breakdown, which is caused due to the surge voltage, isprevented.

Also, the snubber circuit 200 is provided with the (N+1) dischargingpaths 22 that cause the current to flow from the side of thenegative-side terminal 202 toward the side of the positive-side terminal201 via at least one of the negative-side capacitor 213 and thepositive-side capacitor 211. Therefore, when the current is caused toflow by the semiconductor module 5, the energy accumulated in thepositive-side capacitor 211 and the negative-side capacitor 213 isdischarged and the discharging voltage of each discharging path 22 isreduced to the voltage between the positive-side terminal 201 and thenegative-side terminal 202.

Here, when the current is interrupted, the charging voltage of each ofthe N charging paths 21 is higher than the discharging voltage of eachof the discharging paths 22, so that the energy having charged thecharging path 21 as a result of the interruption of current cannotfurther charge the charging path 21 even if the energy is discharged bythe discharging path 22. Therefore, the energy having charged thepositive-side capacitor 211 and the negative-side capacitor 213 when thecurrent is interrupted is accumulated and regenerated by thepositive-side capacitor 211 and the negative-side capacitor 213 withoutbeing charged and discharged and thus consumed as loss of circuit by aresonance operation of the wire inductance 1011 and the positive-sidecapacitor 211 and negative-side capacitor 213. Thereby, the loss ofcircuit due to the resonance operation is reduced.

Also, since the element breakdown due to the surge voltage upon theinterruption of current is prevented and the loss of circuit is reduced,an acceptable amount of inductance of wires connected to thepositive-side terminal 51 and the negative-side terminal 52 of thesemiconductor module 5 can be increased. That is, it is possible toincrease a degree of freedom of wire lengths of the positive-side wire101 and the negative-side wire 102.

In the meantime, as described above, in the snubber circuit 200 ofComparative Example, the charging voltage of each charging path 21 whenthe current is interrupted is 4E/3(V). Therefore, a voltage ΔV1, whichis generated due to the wire inductance 1012, of the surge voltageinstantaneously generated between the positive-side wire 101 and thenegative-side wire 102 is generated in a form of being added to 4E/3(V),based on 4E/3(V).

In contrast, when a single snubber capacitor is connected between thepositive-side wire 101 and the negative-side wire 102, a chargingvoltage of the snubber capacitor becomes E(V). Therefore, the voltageΔV1, which is generated due to the wire inductance 1012, of the surgevoltage is generated in a form of being added to E(V). Therefore, in thesnubber circuit 200 of Comparative Example, the entire surge voltageinstantaneously generated between the positive-side wire 101 and thenegative-side wire 102 due to the wire inductance 1012, i.e., a summedvoltage of the voltage ΔV1 and the voltage corresponding to the basebecomes larger than the case where a single snubber capacitor isconnected between the positive-side wire 101 and the negative-side wire102.

[1.1.1(2). Operations of Snubber Circuit 2 of Present Embodiment]

Subsequently, operations of the snubber circuit 2 of the presentembodiment are described. In the meantime, unless otherwise described,the charging voltages of the positive-side capacitor 211 and thenegative-side capacitor 213 in the snubber circuit 2 may be similar tothe snubber circuit 200.

In the state in which the switching element 11 is in an on state and theswitching element 12 is in an off state, the output current flowsthrough a path of the power supply capacitor 10, the positive-side wire101, the switching element 11 and the power supply output terminal 19.At this time, the output current flows through the wire inductance 1012and energy is accumulated therein. When the switching element 11 isturned on from this state, the current may flow in the snubber circuit2, in aspects of a mode (1) to a mode (5).

FIG. 4 shows flow of current in a mode (1). When the switching element11 is turned off, the output current is commutated and flows from thepower supply capacitor 10 and the positive-side wire 101 into eachcharging path 21. At this time, a charging voltage V_(m) of eachauxiliary capacitor 252 is set lower than a charging voltage V_(n) ofthe negative-side capacitor 213, and is 0(V), for example, in thepresent embodiment. For this reason, the current having flowed into thepositive-side capacitor 211 of the charging path 21 flows toward theauxiliary capacitor 252 without flowing toward the first diode 212 andthe negative-side capacitor 213. Thereby, the current energy of the wireinductance 1012 is absorbed by the charging of the positive-sidecapacitor 211 and the auxiliary capacitor 252.

In this way, in the snubber circuit 2 of the present embodiment, at anearly stage at which the current is interrupted, a series circuit of thepositive-side capacitor 211 and the auxiliary capacitor 252 functions asa charging path (also referred to as ‘bypass charging path’), and thecharging voltage in each bypass charging path is E(V). Therefore, thevoltage ΔV1, which is generated due to the wire inductance 1012, of thesurge voltage is generated in a form of being added to E(V), based onE(V).

In the meantime, the voltage V_(m) of the auxiliary capacitor 252 mayincrease to E/3(V).

Thereby, a summed voltage of the positive-side capacitor 211 in serieswith the negative-side capacitor 213 on the bypass charging path and theauxiliary capacitor 252 increases to a voltage of the positive-sidecapacitor 211 in series on the same charging path 21 as thecorresponding negative-side capacitor 213. As an example, a summedvoltage of the voltage 2E/3(V) of the positive-side capacitor 211 on thebypass charging path including the positive-side capacitor 211 of thecharging path 21(2) and the negative-side capacitor 213 of the chargingpath 21(1) and the voltage V_(m)(V) of the auxiliary capacitor 252increases to the voltage E(V) of the positive-side capacitor 211 of thecharging path 21(1).

In other words, the summed voltage of the negative-side capacitor 213and the auxiliary capacitor 252 on the bypass charging path into whichthe current flows from the positive-side capacitor 211 increases to thevoltage of the negative-side capacitor 213 on the same charging path 21as the positive-side capacitor 211. As an example, a summed voltage ofthe voltage E/3(V) of the negative-side capacitor 213 on the bypasscharging path into which the current flows from the positive-sidecapacitor 211 of the charging path 21(2) and the auxiliary capacitor(V)increases to the voltage 2E/3(V) of the negative-side capacitor 213 ofthe charging path 21(2).

As a result, the voltage between the positive-side wire 101 and thenegative-side wire 102 increases from E(V) to 4E/3(V).

FIG. 5 shows flow of current in a mode (2). When the voltage V_(m) ofthe auxiliary capacitor 252 reaches E/3(V), a potential on theanode-side of the first diode 212 of each charging path 21 becomeshigher than the cathode-side, so that the first diode 212 becomesconductive and the current flowing into the positive-side capacitor 211of the charging path 21 flows through the negative-side capacitor 213having a greater capacity than the auxiliary capacitor 252. Thereby, thecurrent energy of the wire inductance 1012 is absorbed by the chargingof the positive-side capacitor 211 and the negative-side capacitor 213.In the meantime, the current flowing through the positive-side capacitor211 may also flow slightly through the auxiliary capacitor 252 to finelyincrease the voltage V_(m) of the auxiliary capacitor 252. By the aboveconfiguration, the current energy of the wire inductance 1012 iscompletely absorbed by the positive-side capacitor 211 and negative-sidecapacitor 213 and the auxiliary capacitor 252, so that the charging iscompleted. In the meantime, the energy of the current flowing into thepositive-side capacitor 211 in the mode (2) generates a voltage ΔV2 dueto the wire inductance 1011.

FIG. 6 shows flow of current in a mode (3). When the current energy ofthe wire inductance 1012 is completely absorbed and the charging is thuscompleted, the discharging is performed via the bypass charging pathbecause the charging voltage is equal to or higher than 4E/3(V) in thebypass charging path. In the meantime, since the capacity of eachauxiliary capacitor 252 is less than the capacity of each of thepositive-side capacitor 211 and the negative-side capacitor 213, thedischarging may also be mainly performed from the auxiliary capacitor252. Thereby, the voltage of each auxiliary capacitor 252 becomes 0(V),and the voltage between the positive-side wire 101 and the negative-sidewire 102 is reduced from 4E/3(V) to E(V).

FIG. 7 shows flow of current in a mode (4). When the voltage between thepositive-side wire 101 and the negative-side wire 102 is reduced to E(V)and the voltage V_(m) of the auxiliary capacitor 252 becomes 0(V), thecurrent flows out from the snubber circuit 2 due to a self-inducingaction of the wire inductance 1011, so that the discharging is performedfrom the positive-side capacitor 211 and the negative-side capacitor 213of each discharging path 22. Thereby, the voltage between thepositive-side wire 101 and the negative-side wire 102 may be reducedfrom E(V) to (E-AVs).

FIG. 8 shows flow of current in a mode (5). When the voltage between thepositive-side wire 101 and the negative-side wire 102 is reduced to(E-AVs) by the discharging from the positive-side capacitor 211 and thenegative-side capacitor 213, the current from the wire inductance 1011again flows into each charging path 21 due to a difference from a DCelectromotive force E. At this time, the charging voltage V_(m) of eachauxiliary capacitor 252 is lower than the charging voltage V_(n) of thenegative-side capacitor 213, and is 0(V), for example, in the presentembodiment. For this reason, the current flowing into the positive-sidecapacitor 211 of the charging path 21 again flows toward the auxiliarycapacitor 252 without flowing toward the first diode 212 and thenegative-side capacitor 213. Thereby, the current energy of the wireinductance 1011 is once absorbed by the charging of the positive-sidecapacitor 211 and the auxiliary capacitor 252. In the meantime, theenergy of the current caused to flow into the charging path 21 by themode (5) may generate a voltage ΔV3 due to the wire inductance 1012.

Thereafter, the charging and discharging in the mode (2) to the mode (5)are repeated by resonance of the wire inductance 1011, the auxiliarycapacitor 252 and the like, so that the voltage V_(m) of the auxiliarycapacitor 252 converges to substantially ΔVs/2(V). Thereby, thecommutation associated with the turn-off operation of the switchingelement 11 is completed. The energy lost due to the resonance is aboutΔVs/2(V), and may be less than the energy corresponding to E/3(V), forexample.

When the switching element 11 is again turned on, as with ComparativeExample, the energy during the turn-off operation accumulated in thepositive-side capacitor 211 and negative-side capacitor 213 and theauxiliary capacitor 252 is released, so that the commutation associatedwith the turn-on operation of the switching element 11 is completed.Thereby, in the present embodiment, as an example, the charging voltageV_(m) of each auxiliary capacitor 252 may become 0(V).

According to the snubber circuit 2, the auxiliary capacitors 252 areeach connected in parallel to at least one of the plurality of firstdiodes 212 and the plurality of second diodes 221. Therefore, when thevoltage ΔV1 is generated due to the wire inductance 1012, the currentflowing from the positive-side wire 101 to the negative-side wire 102 isdrawn from the positive-side capacitor 211 into the auxiliary capacitor252 and charges the auxiliary capacitor 252. Therefore, the base voltageof the voltage ΔV1 that is generated due to the wire inductance 1012 canbe set to the summed voltage (=E) of the positive-side capacitor 211 andthe auxiliary capacitor 252, not the summed voltage (=4E/3) of thepositive-side capacitor 211 and the negative-side capacitor 213.Therefore, it is possible to reduce the surge voltage that isinstantaneously generated between the positive-side wire 101 and thenegative-side wire 102.

Also, each auxiliary capacitors 252 is connected in parallel to each ofthe (N+1) second diodes 221. Therefore, even when there is a differencein the wire path length of the bypass charging path passing theauxiliary capacitor 252, the energy of the voltage ΔV1 that is generateddue to the wire inductance 1012 can be absorbed at the early stage bythe bypass charging path having the shorter path length, so that thesurge voltage can be securely reduced.

Also, since the capacity of the auxiliary capacitor 252 is less than thecapacity of each positive-side capacitor 211 and the capacity of eachnegative-side capacitor 213, it is possible to reduce the energy lossdue to the resonance of the wire inductance 1011 and the auxiliarycapacitor 252.

Also, since the capacity of the auxiliary capacitor 252 is equal to orgreater than 1/1000 of the capacity of each positive-side capacitor 211and the capacity of each negative-side capacitor 213, it is possible tosecurely reduce the surge voltage by drawing the current, as compared toa case where the capacity of the auxiliary capacitor 252 is less than1/1000. Also, as compared to a case where the capacity of the auxiliarycapacitor 252 is greater than 1/100 of the capacity of eachpositive-side capacitor 211 and the capacity of each negative-sidecapacitor 213, it is possible to securely reduce the energy loss due tothe resonance of the wire inductance 1011 and the auxiliary capacitor252.

Also, since the wire inductance of each charging path 21 is less thanthe wire inductance of each discharging path 22, it is possible tosecurely reduce the surge voltage by the charging path 21. Also,generation of excessive inrush current when the current is caused toflow by the discharging can be prevented by the wire inductance of thedischarging path 22.

2. Operation Waveforms

FIG. 9 shows voltages applied to the switching element 11 when theswitching element 11 is turned off and becomes non-conductive. In FIG.9, the vertical axis indicates a voltage, and the horizontal axisindicates time. Also, in FIG. 9, the left graph indicates an operationwaveform when a single snubber capacitor is connected between thepositive-side wire 101 and the negative-side wire 102. In FIG. 9, thecentral graph indicates an operation waveform when the snubber circuit200 of Comparative Example is connected between the positive-side wire101 and the negative-side wire 102. In FIG. 9, the right graph indicatesan operation waveform when the snubber circuit 2 of the presentembodiment is connected between the positive-side wire 101 and thenegative-side wire 102.

As shown with the left graph in FIG. 9, when the single snubbercapacitor is connected, the voltage ΔV1 caused due to the wireinductance 1012 is generated in a form of being added to the voltageE(V) of the power supply capacitor 10, and the energy of the voltage ΔV2caused due to the wire inductance 1011 is lost by the resonance of thewire inductance 1011 and the snubber capacitor.

Also, as shown with the central graph in FIG. 9, when the snubbercircuit 200 of Comparative Example is connected, the voltage ΔV1 (a peakvalue of the voltage generated in the wire inductance 1012 due to thecommutation to the snubber circuit 200 during the turn-off operation ofthe switching element 11 or 12) is generated in a form of being added tothe voltage 4E/3(V) of the charging path 21. Also, since a resonance ofthe wire inductance 1011 and the snubber capacitor is not generated, theenergy of the voltage ΔV2 is regenerated with being lost.

As shown with the right graph in FIG. 9, when the snubber circuit 2 ofthe present embodiment is connected, unlike the case where the snubbercircuit 200 of Comparative Example is connected, the voltage ΔV1 isgenerated in a form of being added to the voltage E(V) of the powersupply capacitor 10 without being generated in the form of being addedto the voltage 4E/3(V) of the charging path 21, so that the elementbreakdown is prevented. Also, since the capacity of the auxiliarycapacitor 252 resonating with the wire inductance 1011 is small, theenergy of the voltage ΔV2 is regenerated without being substantiallylost.

In the meantime, a broken line part of the right graph may also be thesummed voltage of the positive-side capacitor 211 and the auxiliarycapacitor 252 on the bypass charging path. A slope of the broken linepart may change, in correspondence to the capacity of the auxiliarycapacitor 252. For example, when the capacity of the auxiliary capacitor252 is small, the slope of the broken line part may increase to comeclose to a slope of a solid line graph of a rising part.

Here, a period ΔT1 after the voltage applied to the switching element 11reaches the power supply voltage E until the voltage becomes the summedvoltage of the positive-side capacitor 211 and the negative-sidecapacitor 213 in series with each other and a period ΔT 2 from an end ofthe period ΔT1 to an end of the charging of the positive-side capacitor211 or the negative-side capacitor may satisfy the followingrelationship: ΔT1 is equal to or smaller than ΔT2 and ΔT2 is smallerthan 5×ΔT1. The end timing of the period ΔT1 may also be a timing atwhich the voltage applied to the switching element 11 returns to thesummed voltage (in the present embodiment, as an example, a voltagehaving 4E/3 as an initial value) of the positive-side capacitor 211 andthe negative-side capacitor 213 in series with each other after reachingthe power supply voltage E and then becoming the peak value (E+ΔV1).This timing may also be a timing at which the current energy accumulatedin the wire inductance 1012 becomes zero or a timing at which thecurrent energy is completely absorbed by the snubber circuit 2. The endtiming of the period ΔT2 may also be a timing at which the energy of thewire inductance 1011 is absorbed to reach 0(A) by the series voltage Vof the positive-side capacitor 211 and the auxiliary capacitor 252. Theperiods ΔT1 and ΔT2 may be adjusted by the capacities of thepositive-side capacitor 211, the negative-side capacitor 213 and theauxiliary capacitor 252.

As described above, according to the snubber circuit 2 of the presentembodiment, since the periods ΔT1 and ΔT2 satisfy the relationship inwhich ΔT2 is smaller than 5×ΔT1, the energy accumulated in the auxiliarycapacitor 252 is less, as compared to a case in which ΔT2 is equal to orgreater than 5×ΔT1, so that it is possible to reduce the loss due to theresonance of the wire inductance 1011 and the auxiliary capacitor 252.

3. Modified Embodiments

FIG. 10 shows a power conversion apparatus 1A in accordance with amodified embodiment. A snubber circuit 2A of the power conversionapparatus 1A may include at least one auxiliary capacitor 251 eachconnected in parallel to at least one of the N first diodes 212 includedin the N charging paths 21. In the present modified embodiment, as anexample, the snubber circuit 2 includes the N auxiliary capacitors 251,and each auxiliary capacitor 251 is connected in parallel to each of theN first diodes 212. In a case where the auxiliary capacitor 251 haspositive and negative polarities, each auxiliary capacitor 251 may beconnected at the negative polarity to the anode-side of the first diodeand may be connected at the positive polarity to the cathode-side.

The capacity of each auxiliary capacitor 251 may be less than thecapacity of each positive-side capacitor 211 and the capacity of eachnegative-side capacitor 213. For example, the capacity of the auxiliarycapacitor 251 may be 1/1000 to 1/100 of the capacity of eachpositive-side capacitor 211 and the capacity of each negative-sidecapacitor 213. The capacities of each of the auxiliary capacitors 251may be the same or different each other.

Also, a charging voltage V_(m) of each auxiliary capacitor 251 may belower than a charging voltage V_(n) of the negative-side capacitor 213at a timing at which the switching elements 11 and 12 interrupt current,and may be a negative voltage, for example. In the present modifiedembodiment, as an example, an absolute value of the voltage V_(m) ofeach auxiliary capacitor 251 may be the same as the voltage V_(n) of thenegative-side capacitor 213. Thereby, each auxiliary capacitor 251 candraw the current flowing from the side of the positive-side terminal 201toward the first diode 212.

[3.1. Operations of Snubber Circuit 2A of Modified Embodiment]

Subsequently, operations of the snubber circuit 2A of the presentmodified embodiment are described. In the meantime, unless otherwisedescribed, the charging voltages of the positive-side capacitor 211 andthe negative-side capacitor 213 in the snubber circuit 2A may be similarto the snubber circuits 2 and 200.

In the state in which the switching element 11 is in an on state and theswitching element 12 is in an off state, the output current flowsthrough a path of the power supply capacitor 10, the positive-side wire101, the switching element 11 and the power supply output terminal 19.At this time, the output current flows through the wire inductance 1012and energy is accumulated therein. When the switching element 11 isturned off from this state, the current may flow in the snubber circuit2A, in aspects of a mode (1A) to a mode (5A).

FIG. 11 shows flow of current in a mode (1A). When the switching element11 is turned off, the output current is commutated and flows from thepower supply capacitor 10 and the positive-side wire 101 into eachcharging path 21. At this time, the charging voltage V_(m) of eachauxiliary capacitor 251 is set lower than the charging voltage V_(n) ofthe negative-side capacitor 213, and is −E/3(V), for example, in thepresent modified embodiment. For this reason, the current having flowedinto the positive-side capacitor 211 of the charging path 21 flowsthrough the negative-side capacitor 213 via the auxiliary capacitor 251without flowing through the first diode 212. Thereby, the current energyof the wire inductance 1012 is absorbed by the charging of thepositive-side capacitor 211, the auxiliary capacitor 251 and thenegative-side capacitor 213.

In this way, in the snubber circuit 2A of the present modifiedembodiment, at an early stage at which the current is interrupted, aseries circuit of the positive-side capacitor 211, the auxiliarycapacitor 251 and the negative-side capacitor 213 functions as a bypasscharging path, and the charging voltage in each bypass charging path isE(V). Therefore, the voltage ΔV1, which is generated due to the wireinductance 1012, of the surge voltage is generated in a form of beingadded to E(V), based on E(V).

In the meantime, the voltage V_(m) of the auxiliary capacitor 251 mayincrease to 0(V). Thereby, the voltage between the positive-side wire101 and the negative-side wire 102 increases from E(V) to 4E/3(V).

FIG. 12 shows flow of current in a mode (2A). When the voltage V_(m) ofthe auxiliary capacitor 251 reaches 0(V), the current having flowed intothe positive-side capacitor 211 of the charging path 21 flows throughthe negative-side capacitor 213 via the first diode 212. Thereby, thecurrent energy of the wire inductance 1012 is absorbed by the chargingof the positive-side capacitor 211 and the negative-side capacitor 213.In the meantime, the current flowing through the positive-side capacitor211 may also flow slightly through the auxiliary capacitor 251 to finelyincrease the voltage V_(m) of the auxiliary capacitor 251. By the aboveconfiguration, the current energy of the wire inductance 1012 iscompletely absorbed by the positive-side capacitor 211 and negative-sidecapacitor 213 and the auxiliary capacitor 251, so that the charging iscompleted. In the meantime, the energy of the current flowing into thepositive-side capacitor 211 in the mode (2A) generates a voltage ΔV2 dueto the wire inductance 1011.

FIG. 13 shows flow of current in a mode (3A). When the current energy ofthe wire inductance 1012 is completely absorbed and the charging is thuscompleted, the discharging is performed via the bypass charging pathbecause the charging voltage is equal to or higher than 4E/3(V) in thebypass charging path. In the meantime, since the capacity of eachauxiliary capacitor 251 is less than the capacity of each of thepositive-side capacitor 211 and the negative-side capacitor 213, thedischarging may also be mainly performed from the auxiliary capacitor251. Thereby, the voltage V_(m) of each auxiliary capacitor 251 becomes−E/3(V), and the voltage between the positive-side wire 101 and thenegative-side wire 102 is reduced from 4E/3(V) to E(V).

FIG. 14 shows flow of current in a mode (4A). When the voltage betweenthe positive-side wire 101 and the negative-side wire 102 is reduced toE(V) and the voltage V_(m) of the auxiliary capacitor 251becomes−E/3(V), the current flows out from the snubber circuit 2 due toa self-inducing action of the wire inductance 1011, so that thedischarging is performed from the positive-side capacitor 211 and thenegative-side capacitor 213 of each discharging path 22. Thereby, thevoltage between the positive-side wire 101 and the negative-side wire102 may be reduced from E(V) to (E-AVs).

FIG. 15 shows flow of current in a mode (5). When the voltage betweenthe positive-side wire 101 and the negative-side wire 102 is reduced to(E-AVs) by the discharging from the positive-side capacitor 211 and thenegative-side capacitor 213, the current from the wire inductance 1011again flows into each charging path 21 due to a difference from the DCelectromotive force E. At this time, the charging voltage V_(m) of eachauxiliary capacitor 251 is lower than the charging voltage V_(n) of thenegative-side capacitor 213, and is −E/3(V), for example, in the presentembodiment. For this reason, the current flowing into the positive-sidecapacitor 211 of the charging path 21 again flows through thenegative-side capacitor 213 via the auxiliary capacitor 251 withoutflowing through the first diode 212. Thereby, the current energy of thewire inductance 1011 is once absorbed by the charging of thepositive-side capacitor 211 and the auxiliary capacitor 251.

Thereafter, the charging and discharging in the mode (2A) to the mode(5A) are repeated by resonance of the wire inductance 1011, theauxiliary capacitor 251 and the like, so that the voltage V_(m) of theauxiliary capacitor 251 converges to substantially ΔVs/2(V). Thereby,the commutation associated with the turn-off operation of the switchingelement 11 is completed. The energy lost due to the resonance is aboutΔVs/2(V), and may be less than the energy corresponding to E/3(V), forexample.

When the switching element 11 is again turned on, as with ComparativeExample, the energy during the turn-off operation accumulated in thepositive-side capacitor 211 and negative-side capacitor 213 and theauxiliary capacitor 251 is released, so that the commutation associatedwith the turn-on operation of the switching element 11 is completed.Thereby, in the present modified embodiment, as an example, the chargingvoltage V_(m) of each auxiliary capacitor 251 may become −E/3(V).

As described above, also in the snubber circuit 2A of the modifiedembodiment, it is possible to achieve the similar effects to the snubbercircuit 2 of the embodiment.

4. Other Modified Embodiment

In the meantime, according to the embodiment and the modifiedembodiment, the snubber circuit 2; 2A has been described to have eitherthe auxiliary capacitor 252 or the auxiliary capacitor 251 but may haveboth the auxiliary capacitors.

Also, it has been described that the auxiliary capacitors 252 are eachconnected in parallel to each of the (N+1) second diodes 221 but may beconnected in parallel to only some of the second diodes 221. Also, ithas been described that the auxiliary capacitors 251 are each connectedin parallel to each of the N first diodes 212 but may be connected inparallel to only some of the first diodes 212.

While the embodiments of the present invention have been described, thetechnical scope of the invention is not limited to the above describedembodiments. It is apparent to persons skilled in the art that variousalterations or improvements can be added to the above-describedembodiments. It is also apparent from the scope of the claims that theembodiments added with such alterations or improvements can be includedin the technical scope of the invention.

The operations, procedures, steps, and stages of each process performedby an apparatus, system, program, and method shown in the claims,embodiments, or diagrams can be performed in any order as long as theorder is not indicated by “prior to,” “before,” or the like and as longas the output from a previous process is not used in a later process.Even if the process flow is described using phrases such as “first” or“next” in the claims, embodiments, or diagrams, it does not necessarilymean that the process must be performed in this order.

EXPLANATION OF REFERENCES

1: power conversion apparatus, 2: snubber circuit, 3: switch circuit, 5:semiconductor module, 10: power supply capacitor, 11: switching element,12: switching element, 13: flywheel diode, 14: flywheel diode, 19: powersupply output terminal, 21: charging path, 22: discharging path, 51:positive-side terminal, 52: negative-side terminal, 101: positive-sidewire, 102: negative-side wire, 200: snubber circuit, 201: positive-sideterminal, 202: negative-side terminal, 211: positive-side capacitor,212: first diode, 213: negative-side capacitor, 221: second diode, 251:auxiliary capacitor, 252: auxiliary capacitor, 1011: wire inductance,1012: wire inductance

What is claimed is:
 1. A snubber circuit comprising: N (N: integer equalto or greater than 1) parallel charging paths each having apositive-side capacitor, a first diode, and a negative-side capacitorsequentially connected in series between a positive-side terminal and anegative-side terminal, and configured to cause current to flow from aside of the positive-side terminal toward a side of the negative-sideterminal; (N+1) parallel discharging paths each having a second diodeconnected between the negative-side terminal or the negative-sidecapacitor of a k^(th) charging path (k: integer equal to or greater than0 and smaller than N) of the N charging paths and the positive-sidecapacitor of a (k+1)^(th) charging path of the N charging paths or thepositive-side terminal, and configured to cause current to flow from theside of the negative-side terminal toward the side of the positive-sideterminal via at least one of the negative-side capacitor and thepositive-side capacitor; and at least one auxiliary capacitor each beingconnected in parallel to at least one of the N first diodes included onthe N charging paths and the (N+1) second diodes included on the (N+1)discharging paths.
 2. The snubber circuit according to claim 1, whereina capacity of the auxiliary capacitor is less than a capacity of eachpositive-side capacitor and a capacity of each negative-side capacitor.3. The snubber circuit according to claim 2, wherein the capacity of theauxiliary capacitor is 1/1000 to 1/100 of the capacity of eachpositive-side capacitor and the capacity of each negative-sidecapacitor.
 4. The snubber circuit according to claim 1, wherein eachauxiliary capacitor is connected in parallel to any one of the firstdiodes and the second diodes.
 5. The snubber circuit according to claim1, wherein each auxiliary capacitor is connected in parallel to any oneof each of the N first diodes and each of the (N+1) second diodes. 6.The snubber circuit according to claim 1, wherein a wire inductance ofeach charging path is less than a wire inductance of each dischargingpath.
 7. A power conversion apparatus comprising: the snubber circuitaccording to claim 1; and a switch circuit connected to thepositive-side terminal and the negative-side terminal.
 8. The powerconversion apparatus according to claim 7, wherein the switch circuit isan inverter having upper and lower arms, and when any one of the upperand lower arms becomes non-conductive, a period ΔT1 after a voltageapplied to the arm reaches a power supply voltage until the voltagebecomes a summed voltage of the positive-side capacitor and thenegative-side capacitor in series with each other and a period ΔT 2 froman end of the period ΔT1 to an end of charging of at least one of thepositive-side capacitor and the negative-side capacitor satisfy afollowing relationship: ΔT1 is equal to or smaller than ΔT2 and ΔT2 issmaller than 5×ΔT1.